The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

(arithmetic) How many digits does 5437682*34567254 have?

A. 13
B. 14
C. 15
D. 16
E. 17

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Assume x = 5437682 and y = 34567254.
We have 5.4 * 10^6 < x < 5.5 * 10^6 and 3.4 * 10^7 < y < 3.5 * 10^7.
Then we have (5.4 * 10^6)(3.4 * 10^7)< xy < (5.5 * 10^6)(3.5 * 10^7) or 18.36*10^{13} < xy < 19.25*10^{13}, which is equivalent to 1.836*10^{14} < xy < 1.925*10^{14}.
So, xy has 15 digits.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(function) What is the value of f(3)+f(2)?

1) f(a)f(b) = 3f(a+b)+f(a-b)
2) f(1) = 5

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have many variables to determine a function and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
If a = 1, and b = 0, then we have f(1)f(0) = 3f(1+0) + f(1-0) = 4f(1) = 4(5) = 20 or 5f(0) = 20. So, we have f(0) = 4.
If a = 1, and b = 1, then we have f(1)f(1) = 3f(1+1) + f(1-1) = 3f(2) + f(0) = 3f(2) + 4. Therefore 3f(2) + 4 = 25, 3f(2) = 21, or f(2) = 7.
If a = 2, b = 1, then we have f(2)f(1) = 3f(2+1) + f(2-1) = 3f(3) + f(1) = 3f(3) + 5. Therefore 3f(3) + 5 = 5*7, 3f(3) + 5 = 35, 3f(3) = 30, or f(3) = 10.
Then, f(3)+f(2) = 10 + 7 = 17.

Therefore, C is the answer.
Answer: C

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions when the answer is A, B, C, or D.

[GMAT math practice question]

(statistics) In a basketball game, Watson, a new player, substitutes in for James. What is the height of James?

1) The height of Watson is 192 cm.
2) After the substitution, the average height of the 5 players increased by 1.8 cm.

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Assume j is James’ height, and w is Watson’s height.

Since we have 2 variables (j and w) and 0 equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
We have w = 192 and ( w – j ) / 5 = 1.8 which is the increased height after the substitution.
We have w – j =9 and j = w – 9, j = 192 – 9, and j = 183.
Since both conditions together yield a sufficient condition, C appears to be the solution.

Since this question is a statistics question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
We have w = 192, but condition 1) does not give us any information about w.
Condition 1) is obviously not sufficient.

Condition 2)
We have j = w – 9.
Since condition 2) does not yield a unique solution, it is not sufficient.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[GMAT math practice question]

(algebra) What is the length of BE?

1) AB = 24cm and C is the midpoint of AB
2) AD+CE = 5, AD = CD/3



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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 4 variables (AD, DC, CE, and EB) and 0 equations and each condition has 2 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
We have AD + DC + CE + EB = 24 and CE + EB = 12 from condition 1).
Since AD + CD = 12 and AD = CD / 3, we have CD/3 + CD = 12, 4CD/3 = 12, 4CD = 36, or CD = 9. Then AD + 9 = 12, or AD = 3.
Then we have CE = 5 – AD, CE = 5 - 3, or CD = 2.
So, EB = 12 – CE, EB = 12 – 2, or EB = 10.

Since both conditions yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: C

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions when the answer is A, B, C, or D.

[GMAT math practice question]

(algebra) For a positive integer n, f(n) is defined as 1 + 1/2 + 1/3 + … + 1/n. What is the value of 10+f(1)+f(2)+…+f(9)?

A. f(11)
B. 9f(9)
C. 10f(10)
D. f(10)
E. 22

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10 + f(1) + f(2) + … + f(9)
= 10 + ( 1 ) + ( 1 + 1/2 ) + ( 1 + 1/2 + 1/3 ) + … + ( 1 + 1/2 + 1/3 + … + 1/9 )
= 10 + 1*9 + (1/2)*8 + (1/3)*7 + … + (1/8)*2 + (1/9)
= 10 + 1*(10 – 1) + (1/2)*(10-2) + (1/3)*(10-3) + … + (1/8)*(10-8) + (1/9)(10-9)
= 10 + (-1) + (-1) + … + (-1) + 10( 1 + 1/2 + 1/3 + … + 1/9 )
= 1 + 10( 1 + 1/2 + 1/3 + … + 1/9 )
= 10( 1 + 1/2 + 1/3 + … + 1/9 + 1/10)
= f(10)

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(function) f(x) is a function. What is the value of f(2006)?

1) f(11)=11
2) f(x+3)=(f(x)-1)/(f(x)+1)

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have many variables to determine a function and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)

Since we have f(x+3) = (f(x)-1)/(f(x)+1) and f(11)=11, we have f(14)=(f(11)-1)/(f(11)+1)=10/12=5/6 when we substitute 11 for x.
We have f(17) = (f(14)-1)/(f(14)+1) = ((5/6)-1)/(5/6)+1) = (-(1/6))/(11/6) = -1/11 when we substitute 14 for x.
We have f(20) = (f(17)-1)/(f(17)+1) = (-(1/11)-1)/(-(1/11)+1) = (-(12/11))/(10/11) = -12/10 = -6/5, when we substitute 17 for x.
We have f(23) = (f(20)-1)/(f(20)+1) = (-(6/5)-1)/(-(6/5)+1) = (-(11/5))/(-(1/5))=11, when we substitute 20 for x.
Then we have the following patterns.
f(11) = f(23) = f(35) = … = f(12k-1) = 11
f(14) = f(26) = f(38) = … = f(12k+2) = 5/6
f(17) = f(29) = f(41) = … = f(12k+5) = -1/11
f(20) = f(32) = f(44) = … = f(12k+8) = -6/5
So, f(2006) = f(12*167+2) = 5/6.

Therefore, C is the answer.
Answer: C

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions when the answer is A, B, C, or D.

[GMAT math practice question]

(Number) What is a positive integer p?

1) p is a prime number
2) p^2+2 is a prime number

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 1 variable (p) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Since we have an infinite number of prime numbers, we don’t have a unique value of p, and condition 1) is not sufficient.

Condition 2)
If p has a remainder 1 when it is divided by 3 or p=3k+1 for some integer k, then p^2+2 = (3k+1)^2+2 = 9k^2+6k+1+2 = 3(3k^2+2k+1) is a multiple and it is a prime number. We have 3k^2+2k+1=1, 3k^2+2k=0, k(3k+2)=0 and k=0 or k=-2/3. However, k is an integer so only k=0 works. Then p=3(0)+1 = 1. However, p = 1 is not a solution since 1 is not a prime number.

If p has a remainder 2 when it is divided by 3 or p=3k+2 for some integer k, then p^2+2 = (3k+2)^2+2 = 9k^2+12k+4+2 = 3(3k^2+4k+2) is a multiple and it is a prime number. Since we have 3k^2+4k+2=1, 3k^2+4k+1=0 or (3k+1)(k+1)=0 and we have k =-1 and k=-1/3. However, k must be an integer so then p=3(-1)+2 = -1. However, p = -1 is not a solution since -1 is negative.

Assume p has a remainder 0 when it is divided by 3.
If p=3, then p^2+2=11 is a prime number.
If p=9, then p^2+2=83 is a prime number.
Since condition 2) does not yield a unique solution, it is not sufficient.

Conditions 1) & 2)
p is a multiple of 3 from condition 2), and p is a prime number from condition 1). Then p = 3.
Since both conditions together yield a unique solution, it is sufficient.

Therefore, C is the answer.
Answer: C

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations,” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

[GMAT math practice question]

(number properties) m and n are integers. What is the value (-1)^{m-n} +(-1)^{m+n} +(-1)^{mn} +(-1)^{2n}?

1) m = n + 1
2) m = 3

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 2 variables (m and n) and 0 equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Since we have m = n + 1 and m = 3, we can substitute m = 3 into m = n + 1 to get 3 = n + 1 and n = 2.
(-1)^{m-n} +(-1)^{m+n} +(-1)^{mn} +(-1)^{2n}
=(-1)^{3-2} +(-1)^{3+2} +(-1)^{3*2} +(-1)^{2*2}
=(-1)^1 +(-1)^5 +(-1)^6 +(-1)^4
=(-1) + (-1) + 1 + 1 = 0
Since both conditions together yield a unique solution, they are sufficient.

Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
Since m = n + 1, m and n are consecutive integers, they have different parities, which means that if m is an odd integer, then n is an even integer, and if m is an even integer, then n is an odd integer.

Case 1: m is an odd integer and n is an even integer.
Then, m+n is an odd integer, m – n is an odd integer, mn is an even integer and 2n is an even integer.
(-1)^{m-n} +(-1)^{m+n} +(-1)^{mn} +(-1)^{2n}
=(-1)^{odd} +(-1)^{odd} +(-1)^{even} +(-1)^{even}
=(-1) + (-1) + 1 + 1 = 0

Case 2: m is an even integer and n is an odd integer.
Then, m+n is an odd integer, m – n is an odd integer, mn is an even integer and 2n is an even integer.
(-1)^{m-n} +(-1)^{m+n} +(-1)^{mn} +(-1)^{2n}
=(-1)^{odd} +(-1)^{odd} +(-1)^{even} +(-1)^{even}
=(-1) + (-1) + 1 + 1 = 0

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)

Case 1: n is an even integer.
Then, m+n is an odd integer, m – n is an odd integer, mn is an even integer and 2n is an even integer, since m = 3.
(-1)^{m-n} +(-1)^{m+n} +(-1)^{mn} +(-1)^{2n}
=(-1)^{odd} +(-1)^{odd} +(-1)^{even} +(-1)^{even}
=(-1) + (-1) + 1 + 1 = 0

Case 2: n is an odd integer.
Then, m+n is an even integer, m – n is an even integer, mn is an odd integer and 2n is an even integer.
(-1)^{m-n} +(-1)^{m+n} +(-1)^{mn} +(-1)^{2n}
=(-1)^{even} +(-1)^{even} +(-1)^{odd} +(-1)^{even}
=1 + 1 + (-1) + 1 = 2

Since condition 2) does not yield a unique solution, it is not sufficient.

If the question has both C and A as its answer, then A is an answer rather than C by the definition of DS questions. Also, this question is a 50/51 level question and can be solved by using the Variable Approach and the relationship between Common Mistake Type 3 and 4 (A or B).

Therefore, A is the answer.
Answer: A

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[GMAT math practice question]

(Geometry) The figure shows that line m is parallel to the line n, and l is parallel to k. Moreover, ∠CBD=45°, ∠FAE=80°. What is ∠BDC?



A. 40° B. 45° C. 50° D. 55° E. 60°

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Since lines m and n are parallel, we have <FBG=<FAE=80° and 80°+<ABD+45°=180°.
Then we have 125°+<ABD=180 and <ABD=55°.
Since <ABD and <BDC are alternate interior angles, they are congruent and <BDC=55°.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(number properties) p and q are integers. Is (p-1)(q-1) an even number?

1) p+q is an odd number
2) pq is an even number

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

The following reasoning shows that in the question, either p or q is an odd integer.
(p-1)(q-1) is an even integer
=> p – 1 or q – 1 is an even integer
=> p or q is an odd integer

Therefore, either p and q is an odd number, and the other one is an even number, according to condition 1. So, condition 1) is sufficient.

Condition 2)

If p is an odd number and q is an even number, then p-1 is an even number, q-1 is an odd number, and (p-1)(q-1) is an even number, which means the answer is ‘yes’.
If both p and q are even numbers, then (p-1)(q-1) is an odd number, and the answer is ‘no’ since both p-1 and q-1 are odd numbers.

Since condition 2) does not yield a unique solution, it is not sufficient.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(geometry) The figure shows that OA = 20, OB = 30 and OC = x and □OCDE is a rectangle. What is the area of rectangle OCDE?

1) x = 10
2) OE = 15



=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Since the triangle OAB and the triangle CAD are similar, we have OA:OB = 2:3 and CA:CD = 2:3. Then we have 3CA = 2CD or CD = (3/2)(20-x).
So the area of the rectangle OCDE is x*(3/2)(20-x). Therefore, we have one variable in this question.

Since we have 1 variable (x) and 0 equations, D is the most likely answer. So, we should consider each condition separately first.

Condition 1) is sufficient, since it yields a unique solution.


Condition 2)
Since CD = OE = 15 from condition 2), and from the original condition we know CD = (3/2)(20-x).
=>15 = (3/2)(20-x)
=>10 = 20-x
=>x = 10
=>3CA = 2CD
=>3CA = 2(15)
=>3CA = 30
=>CA = 10
We have CA = 10 and x = 10.
So, condition 2) is also sufficient, because it is equivalent to condition 1).

Therefore, D is the answer.
Answer: D

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

[GMAT math practice question]

(propotional) What is z^2/xy + x^2/yz + y^2/zx ?

1) x:y = 2:3
2) x:z = 1:2

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 3 variables (x, y, and z) and 0 equations, E is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Since x:y = 2:3 and x:z = 1:2, we have x:y:z = 2:3:4.
Then we have x = 2k, y = 3k, and z = 4k for some number k.
z^2/xy + x^2/yz + y^2/zx
= (4k)^2/(2k)(3k) + (2k)^2/(3k)(4k) + (3k)^2/(4k)(2k)
= 16k^2/6k^2 + 4k^2/12k^2 + 9k^2/8k^2
= 16/6 + 4/12 + 9/8
= 64/24 + 8/24 + 27/24 = 99/24 = 33/8.

Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: C

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.

[GMAT math practice question]

(equation) What are the values of x+y and xy?

1) x + y + xy = -2
2) (1/x) + (1/y) = 1

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 2 variables (x and y) and 0 equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Since 1/x + 1/y = 1 from condition 2), we have y + x = xy by multiplying both sides of the equation by xy, which rearranges to get xy – (x+y) = 0.
Since xy + (x+y) = -2 from condition 1), we have xy - (x+y) + xy + (x+y) = 0 + -2 by adding the two equations. Then 2xy = -2 or xy = -1.
Then we have x+y=-1.

Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[GMAT math practice question]

(geometry) The figure shows the rectangle ABCD. What is ∠x - ∠y?



A. 20°
B. 17°
C. 15°
D. 13°
E. 12°

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Since AP, RP and CD are parallel, we have <ABP = <BPR and <DQP = <RPQ. Since <BPQ = <BPR + <RPQ, we have <x = 15° + <y.
So, we have <x - <y = 15°.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(algebra) What is the value of x/(x+y) + y/(x-y)?

1) (x+y):y = 3:1
2) x + y = 8

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

The question x/(x+y) + y/(x-y) is equivalent to (x^2+y^2)/(x^2-y^2) for the following reason
x/(x+y) + y/(x-y)
=> x(x-y)/(x+y)(x-y) + y(x+y)/(x+y)(x-y)
=> (x^2-xy+xy+y^2)/(x^2-y^2)
=> (x^2+y^2)/(x^2-y^2)
=> (x^2/y^2+1)/(x^2/y^2-1) by dividing the top and bottom by y^2
=> [(x/y)^2+1)/[(x/y)^2-1]

When a question asks for a ratio, if one condition includes a ratio and the other condition just gives a number, the condition including the ratio is most likely to be sufficient. This tells us that A is most likely to be the answer to this question.

Condition 1)
The condition (x+y):y = 3:1 is equivalent to x = 2y since x + y = 3y from (x+y):y = 3:1.
Then (x^2+y^2)/(x^2-y^2) = ((2y)^2+y^2)/((2y)^2-y^2) = (4y^2+y^2)/(4y^2-y^2) = 5y^2/3y^2 = 5/3.
Since condition 1) yields a unique solution, it is sufficient.


Condition 2)
If x = 5 and y = 3, then we have x/(x+y) + y/(x-y) = 5/8 + 3/2 = 5/8 + 12/8 = 17/8.
If x = 6 and y = 2, then we have x/(x+y) + y/(x-y) = 6/8 + 2/4 = 3/4 + 2/4 = 5/4.
Since condition 2) does not yield a unique solution, it is not sufficient.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(algebra) max{x, y} denotes the maximum of x and y, and min{x, y} denotes the minimum of x and y. What is the value of x + y?

1) max{x, y} = x + y
2) min{x, y} = 2x + y - 2

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 2 variables (x and y) and 0 equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Case 1: x ≥ y
Since max(x,y) = x and max(x,y) = x + y, we have x = x + y or y = 0.
Since min(x,y) = y and min(x,y) = 2x + y - 2, we have y = 2x + y - 2, 0 = 2x - 2, 2x = 2, or x = 1.
Then we have x + y = 0 + 1 = 1.
Case 2: x < y
Since max(x,y) = y and max(x,y) = x + y, we have y = x + y or x = 0.
Since min(x,y) = x, min(x,y) = 2x + y - 2 and x = 0, we have x = 2x + y - 2 or y = 2.
Then we have x + y = 0 + 2 = 2.

Since both conditions together do not yield a unique solution, they are not sufficient.

Therefore, E is the answer.
Answer: E

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[GMAT math practice question]

(equation) What is the value of a + b?

1) ax + by = 2(ax - by) - 3 = x + y + 7
2) x = 3, y = 1

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

We have 4 variables (a, b, x and y). However, since both conditions have 4 equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Since x = 3 and y = 1, we have 3a + b = 2(3a - b) - 3 = 3+1+7 = 11.
Then we have 3a + b = 11 and 6a - 2b = 14 or 3a – b = 7.
When we add those equations we have 3a + b + 3a - b = 11 + 7, 6a = 18 or a = 3.
Then we have 3(3) + b = 11, 9 + b = 11 or b = 2.

Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.